Fake Lie groups and maximal tori. I
[Part II is reviewed below (see Zbl 0697.55007).] A finite loop space X is called a fake Lie group of type G, where G is a compact connected Lie group, if the classifying spaces BX and BG have the same genus. (Recall that two nilpotent spaces of finite type \(Y'\), \(Y''\) are said to have the same genus if their p-localizations are homotopy equivalent for all primes.) Using Rector's idea of a maximal torus for a finite loop space we prove that a fake Lie group is genuine iff it has a maximal torus. For each maximal torus for X there is a Weyl group. We prove that these all coincide in a strong sense with the classical Weyl group of G. Moreover we show that the homotopy fibre of a maximal torus f: BT\(\to BX\) is homotopy equivalent to G/T. Finally we show that a fake Lie group X has a maximal torus iff \(\tilde K(BX)\cong \tilde K(BG)\) as \(\lambda\)-rings.
scientific article; zbMATH DE number 4143031
scientific article